Errors in Nakahara's book

Geometry, Topology and Physics by Mikio Nakahara can be downloaded from the following online page.

(GradMikio Nakahara-Geometry, topology, and physics-Institute of Physics Publishing.pdf
http://staff.ustc.edu.cn/~hyx/0911/(GradMikio%20Nakahara-Geometry,%20topology,%20and%20physics-Institute%20of%20Physics%20Publishing.pdf

If you want to buy the published version, there is an Amazon page and you can get it anytime.

Amazon | Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics) | Nakahara, Mikio | Differential Geometry
https://www.amazon.co.jp/-/en/Mikio-Nakahara/dp/0750306068

1. Example 9.2.

Right after the equation (9.19), there is the following sentence:

Let $W=W^{\nu }\partial /\partial y^{\nu }$ be a vector of $T _p N$ and $V=V^{\mu}\partial /\partial x^{\mu}$ be the corresponding vector of $T _{f ( p)} M$ .

This is wrong, although physicists and physics students don't care about such a thing.

Two vector fields $W$ and $V$ must be actually just vectors, that is,

$W=W^{\nu}\left( \dfrac{\partial }{\partial y^{\nu }}\right) _{p}$

and

$V=V^{\mu }\left( \dfrac{\partial }{\partial x^{\mu}}\right) _{f\left( p\right) }$ .

Obviously, the author writes "a vector of $T _p N$ " so $W$ is not a vector field. Similarly, $V$ is a vector.

2. Section 9.2

Just below the equation (9.10), the author writes the definition of a section.

He writes the set of sections of a fiber bundle $(E,\pi, M, F)$ as $\Gamma(M,F)$ ,

which is incorrect. A section maps a point on the base space to an element of the total space.

In fact, the following example is the sections on $M$ , $\Gamma(M, TM )$ , where the tangent bundle $TM$ is not a fiber but the total space.

Correctly, the formula is

$\Gamma(M,E)$ .

cf. Section (fiber bundle) - Wikipedia
https://en.wikipedia.org/wiki/Section_(fiber_bundle)

3. Section 10.1.3

Below Figure 10.2 in section 10.1.3, he writes

where we have noted that $\sigma _{i\ast }X\in T_{\sigma _{i}}P$

but this is wrong. That's because $\sigma _{i}$ is not a point on $P$ .

In this section, $P$ is a principal bundle. In other words, given a fiber bundle $(E, \pi, M, F, G)$ , the total space is $E \equiv P$ and being a principal bundle gives $F = G$ by definition.

$\circ$ Projection

In section 9.2.1, a differentiable fiber bundle is defined and the projection is important in this case:

(iv) A surjection $\pi :E\rightarrow M$ called the projection. The inverse image $\pi ^{-1}\left( p\right) =F_{p}\cong F$ is called the fiber at $p$ .

A fiber $F_{p}$ at $p$ is a subset of $E$ and just needed to be diffeomorphic to the typical fiber $F$ . If we rewrite it into a mathematical formula,

$F \cong F_{p} = \pi^{-1} [\lbrace p \rbrace] \subseteq E$ .

$\circ$ Section

Def.

Let $(E,\pi, M,F,G)$ be a fiber bundle. A section (or a cross section) $s$ is defined as

$s:M\rightarrow E$
$\pi \circ s=\mathrm{id}_{M}$
smooth (differentiable fiber bundle) or homeomorphic (topological fiber bundle).

By definition, the value of a section at $p$ , $s(p)$ , is an element of a fiber over $p$ , $F_{p} = \pi^{-1}[\lbrace p \rbrace]$ , i.e.

$s(p) \in F_{p} \cong F$ and $F_{p} \subseteq E$ .

The set of (global) sections on $M$ is denoted by $\Gamma(M,E)$ (not $\Gamma(M,F)$ ).

Here, a local section $\sigma _{i}$ is an element of $\Gamma \left( U_{i},E\right) = \Gamma \left( U_{i},P\right)$ , where $U_{i}$ is an open set of the base space $M$ .